A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stress Yg below which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only when Yg is below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value of Yg there is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

Creeping motion of a sphere through a Bingham plastic

A model has been described for the influence of growth rate and shear stresses in the fermentor upon the morphology of filamentous molds. The main concept of this model is the dynamic equilibrium between growth and breakup of the hyphae. The latter has been approached according to well-known engineering theories for dispersion of physical systems. Experiments to verify the model with a strain of Tenicillium chrysogenum in batch and continuous culture revealed that the length of the mycelial particles increased with increasing' growth rate and decreased with increasing power input per unit mass in the fermentor. Although this was qualitatively in agreement with the presented model, quantitatively the model had to be rejected. Variation of the tensile strength of the hyphae with age and culturing conditions could have been one of the causes of disagreement. Oxygen tension, varied independently from stirrer speed, in the range of 12-300 mm Hg was shown to have no influence upon the morphology. With respect to the question of possibly using high-energy inputs in industrial mold fermentation in order to decrease hyphal length and suspension viscosity, it was concluded that this is of little practical value. A substantial decrease in hyphal length requires an enormousmore » increase in energy input.« less

Influence of engineering variables upon the morphology of filamentous molds

Due to varied flow behavior, the demarcation of hydrodynamic flow regimes is an important task in the design and scale-up of bubble column reactors. This article reviews most hydrodynamic studies performed for flow regime identification in bubble columns. It begins with a brief introduction to various flow regimes. The second section examines experimental methods for measurement of flow regime transition. A few experimental studies are presented in detail, followed by the effect of operating and design conditions on flow regime transition. A table summarizes the reported experimental studies, along with their operating and design conditions and significant conclusions. The next section deals with the current state of transition prediction, and includes purely empirical correlations, semi-empirical models, linear stability theory, and Computational Fluid Dynamics (CFD) based studies.

A Review on Flow Regime Transition in Bubble Columns

The published literature on the liquid-phase mixing in a turbulent flow regime has been critically reviewed and analyzed. Experimental techniques for mixing time have been described together with their relative merits. The effects of the impeller design (blade number, blade angle, blade and disk dimensions, and blade shape), the location of the impeller (off-bottom clearance, distance from the vessel center, i.e., eccentricity), and the vessel size on the liquid-phase mixing have been critically analyzed. The mixing performance dependency on the internals such as baffles (number, dimension, and position) and the draft tube has been presented in detail. Further, an extensive review on the mathematical models proposed for the liquid-phase mixing has been presented, and the utility of the computational fluid dynamics modeling for the mixing optimization has been illustrated. Finally, suggestions have been made for the selection of an energy-efficient impeller−vessel configuration, and directions have been gi...

Liquid-phase mixing in stirred vessels: turbulent flow regime

Fluid mechanics and blending in agitated tanks

To express the degree of mixing in chemical process equipment, a new, generally applicable definition of quality of mixedness is presented by making use of the entropy in information theory. The newly defined quality of mixedness varies from zero for the complete separate state or "piston flow" to values approaching unity for the final mixing state or the complete mixing state. The application of the newly defined quality of mixedness is explained with two types of examples.

A definition of quality of mixedness

The decay process of turbulent swirling flow in an axisymmetric circular pipe is analyzed and a method to estimate the tangential mean velocity distribution is proposed. Two basic new factors, viz. the imaginary inlet position of the swirl decay and the friction velocity Uθf for the swirling flow, were defined in order to investigate the distribution of the intensity of swirl Γ. It became clear that there were characteristic functional relationships between the dimensionless radius r+ and the dimensionless intensity of swirl Γ+ in the forced vortex zone and in the quasi-free vortex zone, respectively, where r and Γ were made dimensionless by Uθf. By making use of these relationships between r+ and Γ+ and the exponential decay process of Γ in the axial direction after determination of the values of empirical coefficients, a calculation procedure for tangential mean velocity in a given condition is proposed.

Turbulent swirling flow in a circular pipe

In the impeller stream in a baffled six-flat-blade agitated cylindrical tank, radial, tangential and axial components of mean and fluctuating velocities were measured with a multi-electrode spherical probe based on electrochemical reaction controlled by diffusional mass transfer rate. The impeller stream flow field is characterized satisfactorily by turbulent statistical values about the velocity such as turbulence intensities, one-point double and triple correlations, and power spectrum.

Turbulence in impeller stream in a stirred vessel

Mixing rate in a stirred vessel.

A new type of viscometer for measuring human blood viscosity is made by using a vacuum glass tube and a capillary needle. This new viscometer needs a volume of only about 5 × 10–6–8 × 10–6 m3 of blood. Simple handling, quick measurement and accurate results in a wide shear rate range are its superior features. Additionally, the whole measuring set with the viscometer is easy to use and to carry, and it is not expensive.By using various standard-viscosity Newtonian liquids, it is confirmed experimentally that this viscometer is sufficiently accurate in the range of less than about 10–2 Pa·s. The viscosity of fresh human blood collected in a hospital is measured by the new viscometer and it is confirmed that the viscosity value obtained is reasonable by comparing it with that by a traditional measuring method. Further, it is shown that the new viscometer can measure the viscosities of other liquids which have complex properties.

Shiro Ito

Papers

A definition of quality of mixedness

Turbulent swirling flow in a circular pipe

Turbulence in impeller stream in a stirred vessel

Mixing rate in a stirred vessel.

Blood Viscometer with Vacuum Glass Suction Tube and Needle